On Krull-Gabriel dimension and Galois coverings

TL;DR

This paper investigates the relationship between the Krull-Gabriel dimensions of a category and its Galois quotient, establishing conditions for their finiteness and applying results to specific algebra classes.

Contribution

It proves the equivalence of finiteness of Krull-Gabriel dimensions for categories and their Galois coverings, and applies this to classify certain selfinjective algebras.

Findings

Krull-Gabriel dimension of R is finite iff that of A is finite

When finite, KG(R) equals KG(A)

No super-decomposable pure-injective modules over certain algebras

Abstract

Assume that $K$ is an algebraically closed field, $R$ a locally support-finite locally bounded $K$-category, $G$ a torsion-free admissible group of $K$-linear automorphisms of $R$ and $A=R/G$. We show that the Krull-Gabriel dimension $KG(R)$ of $R$ is finite if and only if the Krull-Gabriel dimension $KG(A)$ of $A$ is finite. In these cases $KG(R)=KG(A)$. We apply this result to determine the Krull-Gabriel dimension of standard selfinjective algebras of polynomial growth. Finally, we show that there are no super-decomposable pure-injective modules over standard selfinjective algebras of domestic type.